Consecutive integers in high-multiplicity sumsets

2008/06/27 by Vsevolod F. Lev, Lev, Vsevolod F.
Computer Science · Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0806.4580

openalex publication_date 2008/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let k,l≥ 1 and n≥ 3 be integers, and suppose that A₁,...,A_k⊂[0,l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k≥ 2\lceil(l-1)/(n-2)\rceil, then the sumset A₁+...+A_k contains a block of consecutive integers of length k(n-1).

Related

• The Typical Structure of Sets with Small Sumset
• Possible Sizes of Sumsets
• The structure of higher sumsets
• Long arithmetic progressions in sumsets: Thresholds and Bounds
• The typical approximate structure of sets with bounded sumset